Operations with Rational Numbers
Aaliyah spends $\frac{1}{4}$ of her birthday money on a video game. Then she spends $\frac{2}{3}$ of what is left on snacks. What fraction of her money is still in her wallet? To answer it you need to add, subtract, multiply and divide fractions with confidence, even when negatives show up.
Simplifying fractions
Every answer in this page should be in simplest form (also called lowest terms): the numerator and denominator have no common factor except 1. Some problems say irreducible fraction. It means exactly the same thing.
In a survey carried out in your class about favorite social networks, $\frac{12}{30}$ of the students chose a short video platform. Find the irreducible fraction to know…
In a survey carried out in your class about favorite social networks, $\frac{12}{30}$ of the students chose a short video platform. Find the irreducible fraction to know what part of the total this group represents.
Adding and subtracting fractions
A teenager dedicates $\frac{1}{3}$ of his free time in the afternoon to playing sports and $\frac{2}{5}$ to studying math. a) Calculate what fraction of his total free…
A teenager dedicates $\frac{1}{3}$ of his free time in the afternoon to playing sports and $\frac{2}{5}$ to studying math.
- a) Calculate what fraction of his total free time these two activities occupy.
- b) What fraction of time does he have left to play on his console?
To prepare a healthy smoothie, you have mixed $\frac{3}{4}$ of a gallon of oat milk with $\frac{1}{2}$ gallon of orange juice. When serving it, $\frac{1}{8}$ of a gallon…
To prepare a healthy smoothie, you have mixed $\frac{3}{4}$ of a gallon of oat milk with $\frac{1}{2}$ gallon of orange juice. When serving it, $\frac{1}{8}$ of a gallon is spilled. Perform the operation step by step to find out how much mixture is left in the pitcher.
Multiplying and dividing fractions
On a shelf in a comic book store, $\frac{2}{3}$ of the copies are superhero comics. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American…
On a shelf in a comic book store, $\frac{2}{3}$ of the copies are superhero comics. Of those superhero comics, half ($\frac{1}{2}$) belong to a famous American publisher. Calculate the fraction of the total shelf that the comics from that specific publisher represent by performing a multiplication of fractions.
Your cell phone battery has a total capacity of $4500$ mAh. If at this moment the screen indicates that it has exactly $\frac{4}{9}$ of its capacity left: a) How many…
Your cell phone battery has a total capacity of $4500$ mAh. If at this moment the screen indicates that it has exactly $\frac{4}{9}$ of its capacity left:
- a) How many mAh does it have available?
- b) If the power saving mode is activated when $\frac{1}{5}$ of the total is left, how many mAh does that limit represent?
We want to distribute $\frac{3}{2}$ of a pound of jelly beans into small bags. If we put $\frac{1}{10}$ of a pound in each bag: a) How many bags will we be able to fill…
We want to distribute $\frac{3}{2}$ of a pound of jelly beans into small bags. If we put $\frac{1}{10}$ of a pound in each bag:
- a) How many bags will we be able to fill completely?
- b) Perform the operation using division of fractions: $\frac{3}{2} : \frac{1}{10}$
Opposites and reciprocals
Observe the following figure representing a chocolate cake divided into equal portions. Write down the fraction that represents the missing part and calculate the…
Observe the following figure representing a chocolate cake divided into equal portions. Write down the fraction that represents the missing part and calculate the opposite fraction and the reciprocal fraction of that amount.
Order of operations with fractions
Fractions follow the same order of operations as whole numbers.
You may know this as PEMDAS. Remember that multiplication and division share a level, and so do addition and subtraction. Work them left to right. With nested brackets, start with the innermost ones. A long fraction bar also acts like parentheses: do the top and the bottom first, then divide.
Solve the following operation respecting the order of operations: $$\frac{2}{5} \cdot \left( \frac{1}{2} + \frac{1}{3} \right) - \frac{1}{10}$$
Solve the following operation respecting the order of operations:
$$\frac{2}{5} \cdot \left( \frac{1}{2} + \frac{1}{3} \right) - \frac{1}{10}$$Solve the following expression, simplifying the result until you obtain the irreducible fraction: $$\frac{3}{5} - \left[ \frac{1}{2} : \left( \frac{2}{3} - \frac{1}{6}…
Solve the following expression, simplifying the result until you obtain the irreducible fraction:
$$\frac{3}{5} - \left[ \frac{1}{2} : \left( \frac{2}{3} - \frac{1}{6} \right) \right] \cdot \frac{5}{4}$$Calculate the result of the following expression, simplifying the result until you obtain the irreducible fraction: $$ \frac{3}{5} - \left[ \frac{1}{2} + \left(…
Calculate the result of the following expression, simplifying the result until you obtain the irreducible fraction:
$$ \frac{3}{5} - \left[ \frac{1}{2} + \left( \frac{2}{3} \right)^2 \cdot \left( \frac{1}{4} - 1 \right) \right] $$Powers and roots of fractions
Calculate the value of the following operation with negative exponent powers and simplify the result: $$ \left( \frac{3}{2} \right)^{-2} : \left( \frac{1}{2} -…
Calculate the value of the following operation with negative exponent powers and simplify the result:
$$ \left( \frac{3}{2} \right)^{-2} : \left( \frac{1}{2} - \frac{1}{3} \right) + \left( -1 \right)^5 $$Find the value of the following expression taking into account the order of operations and the rules of signs: $$ \frac{\frac{3}{2} - \frac{5}{4}}{\frac{1}{2} \cdot…
Find the value of the following expression taking into account the order of operations and the rules of signs:
$$ \frac{\frac{3}{2} - \frac{5}{4}}{\frac{1}{2} \cdot \left( -3 \right)} + \sqrt{\frac{16}{81}} $$Find the value of $x$ in the following equality of powers with rational numbers, applying the necessary properties to simplify the expression: $$\frac{\left( \frac{2}{3}…
Find the value of $x$ in the following equality of powers with rational numbers, applying the necessary properties to simplify the expression:
$$\frac{\left( \frac{2}{3} \right)^5 \cdot \left( \frac{9}{4} \right)^{-2}}{\left( \frac{4}{9} \right)^{3} : \left( \frac{3}{2} \right)^{-4}} = \left( \frac{2}{3} \right)^x$$Decimals and fractions together
Sometimes a problem mixes decimals and fractions. The safest plan is to turn every decimal into a fraction and work only with fractions.
In a chemistry laboratory, a container holds $2,4$ gallons of an acid solution. For an experiment, $\frac{2}{3}$ of the content is extracted first. Then, $0,5$ gallons…
In a chemistry laboratory, a container holds $2,4$ gallons of an acid solution. For an experiment, $\frac{2}{3}$ of the content is extracted first. Then, $0,5$ gallons of distilled water are added to dilute the mixture. Finally, $\frac{3}{5}$ of the resulting mixture is used.
- a) Perform all operations working exclusively with fractions.
- b) How many gallons of mixture are left in the container at the end? Express the result both as a fraction and as a decimal number.
Calculate the exact value of the following expression, first transforming the decimal numbers into their respective generating fractions and simplifying the intermediate…
Calculate the exact value of the following expression, first transforming the decimal numbers into their respective generating fractions and simplifying the intermediate steps:
$$\left( \frac{3}{5} - 0,\widehat{6} \right)^2 \cdot \left( \frac{1}{3} \right)^{-1} + \sqrt{\frac{25}{9}} : 1,2\widehat{5}$$"Fraction of the remainder" problems
When a problem says a fraction of what is left, that fraction applies to the remainder, not to the whole.
Back to the beginning: Aaliyah still has $\frac{1}{4}$ of her birthday money. The snack fraction $\frac{2}{3}$ looked big, but it was a fraction of only $\frac{3}{4}$ of the money.
Three friends decide to buy a present for a classmate who is moving to another school. The first contributes $\frac{2}{5}$ of the total price, the second contributes…
Three friends decide to buy a present for a classmate who is moving to another school. The first contributes $\frac{2}{5}$ of the total price, the second contributes $\frac{1}{3}$ of the remainder and the third completes the payment with $\$16$.
- a) What fraction of the total represents the amount contributed by the second friend?
- b) What is the total price of the present?
- c) How much money did each one put in?
A content creator has recorded a video for his channel. During editing, he cuts $\frac{1}{6}$ of the total footage due to sound errors. Then, he removes $\frac{2}{5}$ of…
A content creator has recorded a video for his channel. During editing, he cuts $\frac{1}{6}$ of the total footage due to sound errors. Then, he removes $\frac{2}{5}$ of what was left because the pace was too slow. In the end, the edited video has a duration of $18$ minutes.
- a) What fraction of the original footage does the final video of $18$ minutes represent?
- b) How long was the original recording before starting to edit?
In an esports competition, the total prize is distributed as follows: the winning team takes $\frac{4}{7}$ of the total, the runner-up takes $\frac{2}{3}$ of the…
In an esports competition, the total prize is distributed as follows: the winning team takes $\frac{4}{7}$ of the total, the runner-up takes $\frac{2}{3}$ of the remainder, and the third place takes the rest. If the difference between the second and third prizes is $\$3.500$, calculate the total value of the prize and how much money each of the three teams receives.
Finding the whole
Sometimes you know what is left, and you must find the original amount. Work out what fraction of the whole the known amount is, then divide.
When the unknown total also appears inside the amount that changes, call it $T$ and write an equation. Express each part of the problem in terms of $T$, then solve with inverse operations.
Mixtures: take some out, put some in
In a mixture problem, track one ingredient at a time. When you remove some of the mixture, you remove the same fraction of each ingredient. If you pour in a pure ingredient to replace it, the total amount does not change.
A young influencer* is organizing her monthly budget derived from her collaborations. She allocates $\frac{2}{5}$ of her income to the rent of her studio and…
A young influencer is organizing her monthly budget derived from her collaborations. She allocates $\frac{2}{5}$ of her income to the rent of her studio and $\frac{1}{4}$ of the remainder to video editing. If after these expenses she still has $\frac{5}{9}$ of what was left to save and she still has $\$480$ available for personal expenses, what is her total monthly income?
Three partners invest in a technology business. The first contributes $\$12.000$, the second contributes $\frac{2}{5}…
Three partners invest in a technology business. The first contributes $\$12.000$, the second contributes $\frac{2}{5}$ of the total and the third contributes the same amount as the other two together minus $\$18.000$.
- a) What is the total investment of the business?
- b) What fraction of the total does each partner contribute? (Express the fractions in their irreducible form).
In a chemistry laboratory, a tank contains a mixture of alcohol and water. Initially, $\frac{2}{7}$ of the content is alcohol. $12$ gallons of the mixture are extracted…
In a chemistry laboratory, a tank contains a mixture of alcohol and water. Initially, $\frac{2}{7}$ of the content is alcohol. $12$ gallons of the mixture are extracted and replaced with $12$ gallons of pure alcohol. If the mixture now contains $\frac{2}{5}$ of alcohol, how many gallons of liquid are there in total in the tank?
Fractions of areas, units, and times
An architect is designing a play area in a rectangular park. He has decided to divide the land as shown in the following diagram: If the "Grass" area occupies exactly…
An architect is designing a play area in a rectangular park. He has decided to divide the land as shown in the following diagram:
If the "Grass" area occupies exactly half of the land and the "Swings" area is $\frac{3}{10}$ of the total park:
- a) What fraction of the total does the "Sand" area represent?
- b) If the park has a total area of $1200$ $m^2$, how many square feet does the "Sand" area measure?
Observe the design of this modern tile for a cultural center. The outer square has an area of $1$ m$^2$. Different regions have been shaded following a geometric…
Observe the design of this modern tile for a cultural center. The outer square has an area of $1$ m$^2$. Different regions have been shaded following a geometric pattern. Determine what fraction of the total the orange shaded area represents and calculate how many cm$^2$ correspond to it.
A surveillance drone makes a round trip between two bases. On the outward journey, it flies with a tailwind at a constant speed. On the return journey, with a headwind,…
A surveillance drone makes a round trip between two bases. On the outward journey, it flies with a tailwind at a constant speed. On the return journey, with a headwind, its speed is reduced to $\frac{3}{5}$ of the outward speed. If the total flight time was $1$ hour and $20$ minutes, what fraction of the total time did it spend on the return journey? Express the result also in minutes and seconds.